Chapter 9: Rotational Motion
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Transcript of Chapter 9: Rotational Motion
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Chapter 9 Lecture
Chapter 9: Rotational Motion
© 2016 Pearson Education, Inc.
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Goals for Chapter 9
• To study angular velocity and angular acceleration.
• To examine rotation with constant angular acceleration.
• To understand the relationship between linear and angular quantities.
• To determine the kinetic energy of rotation and the moment of inertia.
• To study rotation about a moving axis.
© 2016 Pearson Education, Inc.
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Rotational Motion
Rotational Motion
Rigid body instead of a particleRotational motion about a fixed axisRolling motion (without slipping)
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Concepts of rotational motion3 word dictionary
Distance d angle θVelocity angular velocity Acceleration angular acceleration
Angle θ
How many degrees are in one radian ? (rad is the unit if choice for rotational motion) ratio of two lengths (dimensionless)
Factors of unity or
1 radian is the angle subtended at the center of a circle by an arc with length equal to the radius.
Angular velocity
Other units are;
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We Have a Sign Convention – Figure 9.7
© 2016 Pearson Education, Inc.
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Angular acceleration
Relationship between linear and angular quantities
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Tangential component of acceleration;
For;
Radial component
Magnitude of
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Rotational Motion
Angular QuantitiesKinematical variables to describe the rotational motion:Angular position, velocity and acceleration
)(
)(
)(
2
0 t
0 t
rad/s dtd
t lim
rad/s dtd
t lim
rad Rl
ave
ave
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Rotational Motion
Angular Quantities: VectorKinematical variables to describe the rotational motion:Angular position, velocity and accelerationVector natures
x
y
z
R.-H. Rule
)(
)(
)(
2rad/s k dtdkk
rad/s k dtdkk
rad Rl
ˆˆlimˆ
ˆˆlimˆ
0
0
t
t
t
t
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Rotational Motion
“R” from the Axis (O)
Solid Disk Solid Cylinder
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Rotational Motion
Kinematical Equations for constant angular acceleration
constantNote
ax Conversion
:)(
:
020
2
0
200
-2 (3)
(2)21 (1)
, ,
t
t t
v
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Rotational Motion
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Rotation of a Bicycle Wheel – Figure 9.8
θ at t=3.00 s?
3 complete revolutions with an additional 0.34 rev123º
ω at t=3.00 s?
OR using Δθ
© 2016 Pearson Education, Inc.
See Example 9.2 in your text
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On a merry-go-round, you decide to put your toddler on an animal that will have a small angular velocity. Which animal do you pick?
a) Any animal; they all have the same angular velocity.
b) One close to the hub.c) One close to the rim.
© 2016 Pearson Education, Inc.
Clicker question
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Rotational Motion
vtan= same = r11= r22
Tooth spacing is the same 2r1 N1
2r2 N2
=
2
1= N2
N1
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Example 9-3
Given; ; ;
Now;
Energy in rotational motion and moment of inertia;
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Energy in rotational motion and moment of inertia:
With moment of Inertia;For large it is better faster and not so much heavier
Example: fly wheel.[kg m2]
For general shapes; (c - factors)
Hula hoop = all point on the circumference
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Finding the moment of inertia for common shapes
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Example 9.6: An abstract sculpture
a) For axis BC, disks B and C are on axis;
b) For axis through A perpendicular to the plane;
c) If the object rotates with around the axis through A perpendicular to the plane, what is ?
Note: In rotational motion the moment of inertia depends on the axis of rotation. It is not like a mass a constant parameter of an object
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Problem 9.31:
a) Each mass is at a distance from the axis. (axis is through center)
b) Each mass is m from the axis.along the line AB)
c) Two masses are on the axis and two are from the axis.
(axis is alongCD)
The value of I depends on the location of the axis.
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Rotational Motion
Problem 9.34:
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Which object will win?
All moments of inertia in the previous table can be expressed as; (c - number)
Compare;a) For a thin walled hollow cylinder b) For a solid cylinder etc.
Conservation of energy;
Small bodies wins over large bodies.
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Rotational Motion
9.49. Set Up: Apply Eq. (9.19). For an object that is rolling without slipping we have cm .v R Solve: The fraction of the total kinetic energy that is rotational is
2cm
2 2 2 2 2cm cm cm cm cm
(1/2) 1 1(1/2) (1/2) 1 ( / ) / 1 ( / )
IMv I M I v MR I
(a) 2cm (1/2) , so the above ratio is 1/3I MR
(b) 2cm (2/5)I MR so the above ratio is 2/7.
(c) 2cm (2/3)I MR so the ratio is 2/5.
(d) 2cm (5/8)I MR so the ratio is 5/13
Reflect: The moment of inertia of each object takes the form 2.I MR The ratio of rotational
kinetic energy to total kinetic energy can be written as 1.
1 1/ 1
The ratio increases as
increases.
K=
On a horizontal surface , what fraction of the total kinetic energy is rotational?a) a uniform solid cylinder . b) a uniform sphere c)a thin walled hollow sphere d)a hollow cylinder with outer radius R an inner radius R/2On
Small bodies win over large bodies
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You are preparing your unpowered soapbox vehicle for a soapbox derby down a local hill. You're choosing between solid-rim wheels and wheels that have transparent hollow rims. Which kind will help you win the race?
a) Either kind; it doesn't matter.b) The solid-rim wheels.c) The hollow-rim wheels.
© 2016 Pearson Education, Inc.
Clicker question
hollowsolid
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Conservation of energy in a wellFind the speed v of the bucket and the angular velocity w of the cylinder just as the bucket hits the water
Energy conservation;
Note: free fall velocity for M<<<m; all energy goes to kinetic and not rotation.
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Calculate the angular momentum and kinetic energy of a solid uniform sphere with a radius of 0.12 m and a mass of 14.0 kg if it is rotating at 6.00 rad/s about an axis through its center
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Rotational Motion
Relationship between Linear and Angular Quantities
RRR
Ra
Ra
Ra
RRl
l Rl
222
rad
tan
tan
( (3)
(2)
)(
(1)
)
dd
dd
v
v v
v
tt
t
t like t )(
atan
arad